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Predicting Protein Interactions of Concentrated Globular Protein Solutions Using Colloidal Models Mahlet Asfaw Woldeyes, Cesar Calero-Rubio, Eric M. Furst, and Christopher J. Roberts J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b02183 • Publication Date (Web): 19 Apr 2017 Downloaded from http://pubs.acs.org on April 27, 2017

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The Journal of Physical Chemistry

Predicting Protein Interactions of Concentrated Globular Protein Solutions Using Colloidal Models Mahlet A. Woldeyes‡, Cesar Calero-Rubio‡, Eric M. Furst*, Christopher J. Roberts* ‡These authors contributed equally Department of Chemical and Biomolecular Engineering. University of Delaware, Newark, Delaware 19716, United States

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ABSTRACT Protein interactions of α-chymotrypsinogen A (aCgn) were quantified using light scattering from low to high protein concentrations. Static light scattering (SLS) was used to determine the excess Rayleigh ratio (Rex) and osmotic second virial coefficients (B22) as a function of pH and total ionic strength (TIS). Repulsive (attractive) protein-protein interactions (PPI) were observed at pH 5 (pH 7), with decreasing repulsions (attractions) upon increasing TIS. Simple colloidal potential-of-mean force models (PMF) that account for short-range non-electrostatic attractions and screened electrostatic interactions were used to fit model parameters from data for B22 vs TIS at both pH values. The parameters and PMF models from low-concentration conditions were used as the sole input to transition matrix Monte Carlo simulations to predict high concentration Rex behavior. At conditions where PPI are repulsive to slightly attractive, experimental Rex data at high concentrations could be predicted quantitatively by the simulations. However, accurate predictions were challenging when PPI were strongly attractive due to strong sensitivity to changes in PMF parameter values. Additional simulations with higher-resolution coarse-grained molecular models suggest an approach to qualitatively predict cases when anisotropic surface charge distributions will lead to overall-attractive PPI at low ionic strength, without assumptions regarding electrostatic “patches” or multipole expansions.


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INTRODUCTION Colloidal or “weak” protein-protein interactions (PPI) have been shown to correlate, in some cases, with liquid-liquid phase separation, opalescence, crystallization, aggregation rates, and elevated solution viscosity.1–5 The contributions to PPI include steric repulsions, shortranged attractions (van der Waals interactions and hydration effects), and electrostatic attractions and repulsions.6–8 The balance between these forces and their contribution to net PPI depends on the solution condition and the protein of interest.4,7,9 PPI have been quantified experimentally at only low protein concentrations (c2) for most examples in the literature.1,3,4,9–11 In those cases, electrostatic interactions are the major contributor to PPI at low total ionic strength (TIS), due to the long Debye screening length.6,12–14 Conversely, steric repulsions and short-ranged attractions are the main contributors to PPI at higher TIS values when charge-screening suppresses electrostatic interactions.7,15–18 While the same forces are operative at high c2, the average distance between the surfaces of adjacent protein molecules is necessarily much smaller. In colloidal models, the solvent is treated at a continuum level, and this could be regarded as an unphysical approximation when the average distance between protein surfaces is of the order of only a few water layers.14,19–21 Colloidal models also typically neglect anisotropic surface charge distributions, or treat them approximately with dipole or truncated multipole expansions.14,22–24 Therefore, it is not clear whether the balance between short- and long-ranged contributions to PPI at high c2 can be predicted accurately from only low-c2 behavior. Experimentally, scattering techniques including static and dynamic light scattering (SLS and DLS) and small-angle neutron and X-ray scattering have been used to study PPI in a range of conditions. For practical reasons, most work has focused on measuring the osmotic second


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virial coefficient (B22) or related low-concentration interaction parameters.3,4,7,9,10,25 In some cases, the low-concentration interactions have been used to qualitatively infer the behavior (i.e., interactions,

aggregation

rates,

viscosity,

and

phase

behavior)

at

higher

protein

concentrations.3,4,7,26 Recent work has shown that one can accurately quantify PPI in situ at high concentrations with techniques such as Rayleigh scattering, small-angle neutron or X-ray scattering, and express PPI in terms of the Kirkwood Buff integral or low-angle (“zero-q”) static structure factor if the zero-q limit can be accurately reached in small angle experiments.4,11,27–30 The experimentally determined osmotic second virial coefficient has a molecular origin that relates it to the protein-protein potential of mean force. Previous work has often used a colloidal description of inter-particle interactions to describe protein solutions.7,9,31,32 For conditions near crystallization, coarse-grained (CG), implicit-solvent models for potentials of mean force (PMF) have been used to describe PPI.33,34 Similarly, screened-dipole interaction models have been developed and related to thermodynamic properties and phase behavior of proteins in solution.7,9,14,19,35,36 Although this colloidal description of PMFs describing interactions between proteins is historically employed as a minimalist approach, the validity of using colloidal particle models for protein solutions has been questioned in some cases.7,37 Objections to using minimalist models can arise from the complex nature of protein interactions that is influenced by factors such pH, TIS, co-solutes, protein concentration, sequence and protein three-dimensional structure, and the heterogeneous surface chemistry of proteins. Based in part on such arguments, recent examples raise the question of whether purely colloidal descriptions of PPI are practically useful for capturing or predicting high-concentration properties of protein solutions.7,9,23,38


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The present work further examines the challenge of using simple colloidal models to predict experimental protein interactions (via Rayleigh scattering) from low to high protein concentrations (c2). The excess Rayleigh scattering (Rex) profiles of a model protein, αchymotrypsinogen A (aCgn)8,39,40 are experimentally determined as a function of pH, TIS and c2. Experimental B22 values are used to parameterize canonical colloidal PMF models as a function of TIS and pH, in terms of the well-depth for short-ranged attractions, the effective net charge, and the effective dipole moment of aCgn at a given pH. The model parameterization is done without knowledge of the high-c2 behavior. The experimental high-c2 Rex results are then predicted using the low-c2 PMF as the force field in transition matrix Monte Carlo simulations.7,41 The results are discussed from both qualitative and quantitative perspectives, highlighting strengths and weaknesses of the approach for repulsive and attractive PPI conditions. Finally, a higher-resolution coarse-grained (CG) model is used to rationalize and potentially predict the qualitative change from repulsive to attractive electrostatic PPI, without the need for defining or assuming the existence relevant charged “patches” or other geometric measures of anisotropic surface charge distributions.7,9,22,42

MATERIAL AND METHODS Solution preparation. Sodium phosphate buffer stock solutions were prepared by dissolving sodium phosphate monobasic anhydrous (Fisher Scientific, Fair Lawn, NJ) in distilled, deionized water (Milli-Q resistivity 18.2 MΩ.cm, Millipore, Billerica, MA) to reach 5 mM sodium phosphate, and titrated to pH 7.0 ± 0.05 (termed pH 7 below) using 5 M sodium hydroxide solution (Fisher Scientific). Sodium acetate buffer (40 mM) was prepared by dissolving anhydrous sodium acetate (Fisher Scientific) and glacial acetic acid (Fisher Scientific) in


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distilled, deionized water and titrating to pH 5.0 ± 0.05 (termed pH 5 below). Stock salt buffer solutions for both pH 5 and pH 7 were prepared using the same procedures as above with the gravimetric addition of 10, 50, 100, or 300 mM NaCl (Fisher Scientific). All buffer solutions were filtered and stored at 4 °C prior to use. Bulk protein solutions were prepared from 5X crystallized lyophilized aCgn (Worthington Biochemical, Lakewood, NJ) dissolved in stock buffer solution (pH 7) to an approximate

protein

concentration

of

15

mg/mL.

Stock

solution

of

35

mg/mL

phenylmethlylsulfonyl fluoride (PMSF) was prepared by dissolving PMSF (Fluka Chemical, Ronkonkoma, NY; Sigma-Aldrich, St. Louis, MO) in 100% anhydrous ethanol (Decon Labs, King of Prussia, PA). The bulk aCgn solution (40 ml at ~15 mg/ml) was treated incrementally (in 250 µL aliquots) with 1 mL of 35 mg/ml PMSF (10x PMSF mole excess) in order to deactivate potential proteolytically active residual proteases in the commercial material.43 The resulting protein solutions were then filtered and dialyzed using 10 kDa molecular-weight-cutoff (MWCO) dialysis membrane (Spectr/Por, Spectrum Laboratories, Rancho Dominguez, CA) in the desired buffer with four 12-hr buffer exchanges at 4°C to remove any residual salt impurities from the commercial protein material. Concentrated protein stock solutions (greater than 100 mg/mL) were prepared through membrane centrifugation at ~ 3200 RCF using 10 kDa MWCO Amicon-Ultra centrifugal tubes (Millipore). UV-VIS spectrophotometry (Agilent 8453, Santa Clara, CA) was used to determine the concentration of the protein absorbance at 280 nm using an extinction coefficient of 1.97 mL mg-1 cm-1.40 Lower-concentration protein samples were then prepared by gravimetrically diluting the concentrated protein solution in the desired buffer to obtain protein concentration ranging from 1 to 100 mg/mL.


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Static light scattering. Static light scattering (SLS) experiments were conducted using a Wyatt Technology (Santa Barbara, CA) DAWN HELEOS II instrument with laser wavelength of 658.9 nm at a temperature of 25°C. In SLS, the average scattered intensity at a given angle is measured and used to calculate the excess Rayleigh scattering, represented as Rex, as previously reported.11 Rex is related to PPI by eq 1, where K is the optical constant, Mw,app is the protein apparent molecular weight, Mw is the true protein molecular weight, c2 is the protein concentration, and G22 is protein-protein Kirkwood-Buff integral:11 (1)

.

Similarly, the zero-q limit structure factor (Sq=0) is found by dividing eq 1 by c2 and Mw, with the canonical simplifying assumption that Mw,app ≈ Mw. In this case, Sq=0 = 1+c2G22. 4,7,11 The Kirkwood-Buff (KB) theory of solutions quantifies PPI in terms of the proteinprotein KB integral, G22. It is related to the PMF, or equivalently the average molecular pair correlation function in an osmotic system via the pair distribution function, ̅

,

evaluated in the Grand Canonical ensemble, where w22(r) denotes the orientation-averaged PMF, kB is Boltzmann’s constant, and T is the absolute temperature. The relationship between ̅ or w22(r) and G22 at any given protein concentration (c2) is given by

(2)

.

In the limit of dilute protein concentrations (c2→0), G22 = -2B22, because ̅ and w22(r) no longer depend on protein concentration.7,11 KB theory and the corresponding analysis is applicable at both low and high protein concentrations, and can be used to quantify PPI at high concentrations from SLS data.7,11 A negative (positive) G22 value is equivalent to a value of Sq=0


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below (above) 1, corresponding to net repulsive (attractive) interactions relative to an ideal gas mixture. Correspondingly in dilute solutions, positive (negative) B22 indicates net repulsions (attractions) relative to the ideal case of two non-interacting point particles.4,11,42

Transition Matrix Monte Carlo simulations and simulated Rex values. Transition Matrix Monte Carlo (TMMC) is a Monte Carlo (MC) algorithm that allows one to efficiently compute the equilibrium probability (Π) of the number of protein molecules (N2) in the system of volume (V), for an open (Grand Canonical) ensemble across a broad range of concentrations by using a transition probability matrix.7,44 TMMC efficiently provides the protein concentration (as number density) as a function of protein chemical potential. This can be used to compute the excess Rayleigh scattering behavior by evaluating G22 through the simulated osmotic compressibility using histogram reweighting techniques.7 The probability distribution, Π(N2), can be calculated for an arbitrary choice of protein chemical potential via eq 3, and G22 can be computed by subsequently combining this with eq 4.7,45

(3)

(4) In eq 3 and eq 4, µ2 is the specified protein chemical potential and µ2,ref is the protein chemical potential used to simulate a reference distribution, Π(N2|µref). The brackets in eq 4 indicate an average in the Grand Canonical ensemble using the probability distribution function Π(N2) obtained from eq 3. The subscript 2 is included to make clear that the “particles” of interest are the protein species, as all simulations in this work utilize implicit solvent.7,9,32


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Details of the methodology are the same as in earlier work.7 An initial uniform distribution was used for Π(N2), and the updated distribution was subsequently reconstructed at the end of each cycle until it converged to the actual distribution, with each cycle being defined as 106 MC attempts. A MC attempt consisted of one of the following randomly selected moves: a translation, a rotation (if applicable for a given model) or a molecule insertion or deletion. Regular movements (translations and rotations) represented 30% of the MC attempts, while deletions and insertions represented the other 70%. Temperature was kept constant at 298.15 K. Preliminary simulations were used to find an adequate value of the reference chemical potential, depending on the choice of the interaction model (see below). A box length of 180 nm was used and the simulation box was started with an empty system. The final scaled Rex values were obtained by using eq 1 with a Mw value of 25.7 kDa and assuming that Mw,app and Mw are equal.4,7,11 This might lead to an initial uncertainty as high as 5% depending on the measured Mw,app at low-c2 (see Results and Discussion below).

Colloidal PMF models. TMMC simulations were carried out using colloidal PMF models as the pairwise additive force fields, as is typically done in colloidal simulations14,23,31,46. These were based on a classical spherical model that included a square well potential (usw) to model steric repulsions and short-range attractions (van der Waals interactions and hydration effects), and a screened electrostatic potential (uel). The former is given mathematically in eq 5, while the latter was decomposed into three contributions, as shown in eq 6: monopole-monopole interactions (uqq), monopole-dipole interactions (uqµ) and dipole-dipole interactions (uµµ),

(5)


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(6) The electrostatic interactions in eq 6 were modeled using the equations shown by Bratko et al. (eqs 1-7 in Ref 14) and reproduced in Supporting Information. In eq 5, the parameters of interest are the well width (λswσ), the well depth (εsw), and the effective hard-sphere diameter of the protein (σ). The value of σ was selected to provide an accurate value of the steric contribution to B22 (termed B22,ST) so at to match to the value reported for the all-atom structure of aCgn.42 The resulting value of σ was 4.65 nm. λsw was set to 1.1, giving an effective range of non-electrostatic attractions equal to 0.465 nm. εsw was used as a fitting/tuning parameter (see below). The parameters that dictate the contributions from uel are the effective net charge (Qeff) and the effective dipole moment of the protein (μeff) in solution (see Supporting Information). The values of Qeff and μeff were determined using the procedure below. The formal definition of B22 is given in eq 7a, where w22(r, Ω1, Ω2, c2→0) is the PMF between a pair of proteins before any averaging of the orientation-dependent contributions (cf., eq 8). It is evaluated in the dilute limit (denoted by c2→0), such that multi-protein interactions are irrelevant. As shown in eq 8, this PMF is a function of the inter-protein distance (r), and two orientational degrees of freedom (Ω1, Ω2). However, these two orientational degrees of freedom can be pre-averaged by computing an orientation-averaged Boltzmann factor of the interaction potential, exp[-w22(r, c2→0)/kBT]. In this sense, eq 7a can be reduced to a one dimensional integral with respect to r as shown in eq 7b. This allows one to compute B22 more efficiently using numerical integration to solve the integral in eq 7b. Consequently, an orientation-averaged dipole equation14 was used to compute the low-c2 contribution to PPI (B22). The electrostatic interactions were decoupled into an orientation-independent interaction (monopole-monopole)


10

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and an orientation-averaged interaction (monopole-dipole and dipole-dipole interactions) using the mathematical models proposed by Bratko et. al. (eqs 9-22 in Ref 14 and eqs 8-12 in Supp. Info). The orientationally averaged electrostatic model involves the same adjustable parameters: Qeff and μeff. B22 was subsequently obtained by numerical integration of eq 7b using MatlabTM (Mathworks Inc., Natick, MA).

(7a)

(7b)

(8)

B22 from three-dimensional protein structure and Mayer Sampling with Overlap Sampling. A structurally higher resolution CG model was used to evaluate whether or not the anisotropic distribution of charges would be expected to lead to multipole dominated behavior. The previously developed one-bead-per-amino acid (1bAA) and four-beads-per-amino acid (4bAA) models7,9,32,42,47 were used to compute B22 using the Mayer Sampling method employing the Overlap Sampling algorithm (MSOS) developed by Kofke and coworkers.7,48 An update to the 1bAA force field proposed by Blanco et. al.9 was made here: the electrostatic interaction was updated from a modified Yukawa potential (eq 4 in Ref. 9) to a screened Coulomb potential following Bratko et. al.14 (eqs 1 & 4 in Ref 14 and Supp. Info). This is written in simplified notation and shown in eq 9, where the first term in the numerator (εcc) is used to account for deviations from theoretical net charge due to effects such as preferential counterion binding.9,49–


11


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The combination of the second term in the denominator ([1+(κσij)/2] 2) and the scaling of the

exponent by κ is used to account for the mean-field Debye-Huckel effects of counter-ions around the protein surface as a function of TIS (or κ) of the solution. This was found to better capture the relationship between B22 and TIS for systems that display relatively weak repulsive or attractive interactions. Additionally, εcc can be related to ψi, the ratio between the effective charge (or more accurately, the effective valence) in solution, q , and the idealized value, qi, as shown in eq 10, where σi represents the size of the charged bead and qi is the valence of amino acid i based on the pH and nominal pKa value of side chain i.. The formula for ψi was obtained by solving the screened Coulomb potential for a protein solution (relative permittivity, ε ~ 78.5) with the valence for charged beads, qi and qj, being +1 or -1, for basic and acidic residues, resepctively (see below). For the sake of simplicity, ψi was assumed to be independent of the amino acid identity. The 4bAA model employed the same interaction parameters as in previous work with the same modification to the electrostatic model as discussed above.47,52 (9) (10) In eq 9, uel(rij) represents the electrostatic interaction between bead i on one protein molecule, and bead j on the other protein, with a center-to-center distance between beads denoted as rij. For the 1bAA model, each amino acid is represented as a single bead, and the charge (qi or qj) for a given amino acid resides at the center of that bead. For the 4bAA model each amino acid is represented by 3 backbone beads and one sidechain bead. Charges reside in the center of the side-chain bead for charged amino acids. Based on nominal pKa values,53 at pH 5, all D and E amino acids are approximated as having a charge of -1, while all H, K, and R amino acids have a


12

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charge of +1. For pH 7, the same charge states apply for D, E, K, and R amino acids, while H is neutral. A similar methodology to the one employed by Calero-Rubio et. al.7 was used here: MSOS simulations were performed at constant temperature (298.15 K) with 107 MC attempts for both the reference system and the model of interest. Each MC attempt consisted of either a translation or a rotation around the center of mass of the molecule. The maximum displacement and rotation for a given step was obtained with a pre-equilibration step of 105 MC attempts to obtain an acceptance ratio of 50%. The steric-only behavior of the full protein structure (1bAA or 4bAA) was used as a reference, so the simulation directly returned B22/B22,ST, and no subsequent rescaling was needed. The strength of short-ranged (i.e., van der Waals and hydrophobic) attractions, εsr (formerly denoted as εhp in Ref 9), was set to 0.36 kBT and 0.93 kBT for the 1bAA and 4bAA CG models, respectively, as these were found to fit all B22/B22,ST values for TIS > 300 mM while setting εcc = 0 kBT (i.e. turning off any electrostatic interaction, see also Ref 9 and Supporting Information).9 This value for εsr was held constant for the following simulations: B22/B22,ST was calculated for εcc values between 0 and 5 kBT in increments of 0.25 kBT, and TIS values of 10, 60, 110 and 300 mM for both CG models. The resulting set of B22/B22,ST values were used to build surface response plots for further analysis (see Discussion below). Statistical uncertainties were estimated by performing 5 independent simulations for each model. The standard deviation was used as the estimate of statistical uncertainty, including error propagation.

Average relative deviation (ARD) calculations and model validation. In order to evaluate the effectiveness of colloidal models to fit or predict experimental PPI, the average relative deviation


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(ARD) was calculated for any given data set as shown in eq 11, where n represents the number of data points and xi is the experimental or simulated value to be evaluated (B22 vs TIS, and Rex vs c2, in this work). As the value of ARD is a measure of the average deviation between the model and the experimental data, a cutoff value between 5% and 25% was used below as the criterion for considering a prediction to be quantitatively accurate, as this average deviation can be considered a conservative estimate of the model prediction uncertainty.

(11)

RESULTS AND DISCUSSION Interactions at dilute protein concentrations SLS was used to determine excess Rayleigh scattering (Rex) as a function of c2 for aCgn solutions at a range of solution conditions. Figure 1A shows Rex vs. c2 as a function of pH and NaCl concentration for c2 < 10 mg/mL. The Rex profiles differ qualitatively between pH 5 (main panel) and pH 7 (inset). Increasing TIS by adding NaCl decreases the upward curvature in Rex vs. c2 at pH 7, while it decreases downward curvature at pH 5. As shown in eq 1, upward (downward) curvature for Rex vs. c2 in Figure 1A indicates net attractive (repulsive) interactions. The high TIS conditions at pH 5, and all of the pH 7 conditions, show net attractions, while the lower TIS pH 5 conditions show repulsive interactions. This indicates that screened electrostatic interactions are important at both pH conditions.


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Figure 1. (A) Excess Rayleigh scattering as a function of protein concentration for sodium acetate at pH 5 (main panel) and sodium phosphate at pH 7 (inset), with increasing total ionic strength (TIS) as indicated by the arrows. Lines represents the fits to eq 1. (B) Normalized osmotic second virial coefficient, B22/B22,ST, at pH 7 (red circles) and pH 5 (blue rectangles) as a function of TIS. B22 values were obtained by fitting the data in panel A. Error bars represent 95% confidence levels for the fitted B22 values. The value of B22,ST = 4.9 mL/g was calculated separately from the all-atom PDB structure of aCgn, as per Gruenberger et al.42, and was treated as independent of solution conditions. The data in Figure 1A were used to regress values of Mw,app and G22 from eq 1 for aCgn at each of the solution conditions. Mw,app was not found to be statistically different from Mw of aCgn (25.7 kDa) in all buffer conditions (see Supp. Info). That is, the normalized apparent molecular weights (Mw,app/Mw) were approximately equal to 1, indicating that no measurable aggregation or solvent-solute non-idealities were present.7,11,54 The G22 values were used to calculate B22/B22,ST, using the relationship B22 = -G22/2 and assuming the dilute approximation for these conditions.11 B22,ST = 2/3πσ3 is the steric contribution to the second virial coefficient, with σ defined as the effective protein diameter (= 4.65 nm, see Methods), based the actual B22,ST calculated from the crystal structure.42 On this scale, values of B22/B22,ST larger (less) than 1 indicate net-repulsive (net-attractive) interactions beyond steric repulsions. In the remainder of


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this report, the terms net-repulsive and net-attractive will be used relative to the steric-only value, rather than the value of zero for an ideal gas mixture.7,11,42 The resulting values of B22 are plotted in Figure 1B as a function of TIS. The data in Figure 1B illustrate that the nature of protein-protein interactions is qualitatively different at pH 5 and 7. At low TIS, B22 decreases with increasing pH, consistent with decreasing the net protein charge as pH values approach the pI of the protein. The theoretical net charge of aCgn is +5 and +9 at pH 7 and pH 5, respectively (as calculated using the amino acid sequence for aCgn55 in PROPKA56). At pH 7, B22/B22,ST was lower than 1 for all TIS, indicating net-attractive interactions. Additionally, the B22/B22,ST values increase (become less negative) with increasing ionic strength. This indicates that attractive electrostatic interactions contribute strongly to the net PPI, and this is likely due to dipole or higher multipole interactions that presumably overcome monopole-monopole repulsions.22 Alternatively, at pH 5 the interactions are strongly net-repulsive at low ionic strength, and this changes to slightly net-attractive interactions with increasing ionic strength. This trend is consistent with canonical behavior for screened monopole-monopole interactions between charged colloids that have non-electrostatic shortranged attractions. Overall, the PPI at low c2 and lower TIS for aCgn are dominated by screened Coulomb interactions at pH 5 and 7; with primarily monopole-monopole repulsions at pH 5, and dipole or multipole contributions dominating closer to the isoelectric point (pI) at pH 7. At high ionic strength, both pH 5 and pH 7 conditions result in quantitatively similar values of B22 and show weakly net-attractive PPI (B22/B22,ST near -1).

Modeling colloidal interactions at low protein concentrations


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The orientation-averaged colloidal interaction models described in the Methods section were used to regress model parameters based on the measured B22 values as a function of TIS (Figure 1B). The data for pH 5 indicated predominantly screened monopole-monopole interactions at low TIS. Therefore, the B22 data for pH 5 were fit to an electrostatic model that only included monopole-monopole repulsive interactions (i.e. μeff = 0) along with short-ranged non-electrostatic attractions and steric repulsions. Based on the strongly attractive electrostatic interactions evident in Figure 1B at pH 7, the B22 values versus TIS for those conditions were fit to the screened electrostatic model that included monopole-monopole, monopole-dipole and dipole-dipole interactions. This was done as a first-order approximation to capture the experimental results at both pH values. As noted in the Methods section, the monopole model has two fitted parameters (εsw and Qeff), and the monopole + dipole model has three fitted parameters (εsw, Qeff and μeff).

Figure 2. Comparison of parameter optimization to colloidal models (solid lines) with the experimental results from Figure 1B (symbols), as a function of TIS. (A) pH 5 with minimum ARD line shown at εsw = 1.87 kBT, Qeff = 3.4 and dashed lines indicating range of values obtained from 10% ARD calculations (gray area in inset). (B) pH 7 with εsw = 1.65 kBT, Qeff = 4.2, µeff = 693 D and dashed lines indicating 10% ARD range from inset. Insets show surface plots of ARD for computed and experimental B22 values across the range of TIS, as a function of εsw and Qeff (pH 5) and Qeff and µeff (pH 7).


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Figure 2A shows the best-fit (solid line) for the colloidal model to the pH 5 data, with confidence intervals from a 10% ARD shown as dashed lines. The inset shows the response surface of ARD from the model of B22 vs TIS, relative to the experimental results, as a function of εsw and Qeff (see also, Methods). Figure 2B shows the analogous results for pH 7 with confidence intervals from a 10% ARD as dashed lines, and with the inset showing the response surface of ARD as a function of Qeff and μeff. For the pH 7 response surface, the number of parameters was reduced from three to two by using the B22 values at pH 5 and pH 7 at the highest TIS values to obtain a common value of εsw = 1.7 kBT so as to capture the plateau value of B22 vs TIS. The results in Figure 2 show that the simple colloidal description of aCgn is able to qualitatively and quantitatively capture the experimental B22 behavior. Additionally, the value for the dipole moment (μeff) of aCgn at pH 7 (693 D) calculated from these model fits is semiquantitatively similar to values previously reported by Velev et al.39 (518 D) as well as calculations done using the approach described by Felder et al.57 (553 D). The next subsection addresses the question of whether this interaction model can predict higher-c2 behavior if it is used to extrapolate to higher c2 conditions.

TMMC and simulated Rayleigh scattering at high c2 TMMC was used to test whether the simple colloidal PMF models that were fit at low concentrations could then predict the behavior at high concentrations, specifically Rex vs. c2 beyond the dilute limit. The experimental Rex values at higher concentrations were measured in increments of 10 mg/mL up to c2 = 100 mg/mL at TIS = 20, 30, 70 and 120 mM for pH 5; and TIS =10, 20, 60 and 110 mM for pH 7. Using the parameter values derived from only the experimental results in the dilute limit (i.e., using only B22 values), TMMC simulations were


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performed at the same TIS and pH values listed above. In order to assess the sensitivity of the predictions from TMMC to the choice of model parameters, the simulations were performed across the small parameter space shown in the insets in Figure 2 within an ARD cut-off range of 20% around the global minimum for the best-fit parameters from the low-c2 data. It is not practical to use standard regression algorithms to refine model parameters when the predictions of Rex are based on molecular simulations instead of an analytical model. An alternative approach is to consider response surfaces of ARD as a function of the choice of PMF model parameters in the molecular simulation. ARD is analogous to the root mean squared deviation (RMSD) between the predicted and experimental values in a data set, but without the bias towards very large or close-to-zero experimental values that RMSD can suffer from.58,59 In what follows, the ARD values based on all of the Rex data – i.e., all c2 (low to high) and TIS values – for a given pH and choice of model parameter values are denoted as ARDAll. That is done to determine if a single set of model parameters can predict all of the Rex results from low to high c2 and low to high TIS for a given pH. An alternative approach would be to optimize model parameters at a given TIS and pH, and that will be compared below.

Figure 3. Comparison of experimental Rex profiles at high concentrations with predictions from TMMC simulations at pH 5. (A) Contour plot of ARD between experimental and predicted Rex over all c2 values, including low to high ionic strength. The gray area corresponds to ARD values below 5%. (B) Overlay of experimental results (symbols) and predictions from simulation (lines) for the parameter values in panel A that minimized the overall ARD (εsw = 1.8 kBT, Qeff = 4.2).


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(C) Analogue to panel (B) but using the individual ARD in Supp. Info. Insets for (B) and (C) are the corresponding zero-q limit static structure factor values (Sq=0) vs. c2. The ARDAll response surface for the results at pH 5 are shown in Figure 3A, with the experimental data spanning from low to high c2, and low to high TIS. Comparing the ARD response surface from low-c2 (Figure 2A, inset) with the ARDAll response surface for high-c2 (Figure 3A), it can be seen that the small subset of parameters obtained at low-c2 are able to reasonably predict the high-c2 scattering behavior at pH 5 over a range of TIS. A small subset of parameter values produce an overall ARD below 5%, showing that this simple colloidal model could quantitatively predict high-c2 behavior based on a training set at low-c2 for the netrepulsive conditions (low TIS) and weakly attractive conditions (high TIS). Figure 3B shows the measured and predicted Rex values by using parameters within the minimum ARDAll range from Figure 3A. There are small but systematic deviations between the predicted and experimental Rex values with increasing c2 for the 30 mM TIS conditions (red circles), but otherwise the model quantitatively captures the experimental data. An alternative is to refine separate parameter sets for each TIS, in order to acknowledge that the effective charge in the CG model is a lumped parameter and could change with added NaCl based on preferential salt-protein interactions.2,60 Figure 3C shows the analogue to Figure 3B, but for case where the model parameters for each TIS value are optimized separately, based on the ARD profiles for the low-c2 data for that TIS value (see Supp. Info. for individual-TIS ARD calculations). Figure 3C shows that the simulated results quantitatively match the experimental results, with ARD values as low as 2%, for each of the TIS conditions. The agreement between model predictions and experimental results are slightly improved in Figure 3C when compared to Figure 3B, but at the cost of needing multiple sets of model parameters. Overall, the results in Figure 3 indicate that net-repulsive and weakly attractive interactions from


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low to high protein concentration can be captured accurately by simple CG and colloidal models, as suggested in previous work.9

Figure 4. Comparison of experimental Rex profiles at high concentrations and predictions from TMMC simulations at pH 7. (A) Contour plot of ARD between experimental and predicted Rex over all c2 values, including low to high ionic strength. (B) Contour plot of ARD between experimental and predicted Rex over all c2 values, excluding the lowest ionic strength (10 mM). (C) Overlay of experimental results (symbols) and predictions from simulation (lines) for the parameter values in panel A that minimized the overall ARD from panel A (εsw = 1.7 kBT, Qeff = 4.2, μeff = 630 D). (D) Analogue to panel (C) but using the individual ARD plots from Supp. Info. Insets for (C) and (D) are the corresponding zero-q limit static structure factor, Sq=0, vs. c2. The gray area in panel B corresponds to ARD values below 10%. The same methodology was applied to the pH 7 conditions, and the ARDAll response surface is shown in Figure 4A. In this case, there are two distinct regions for the predictions: a


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liquid-liquid (L-L) split regime at high μeff/Qeff values, and a single phase regime for low μeff/Qeff values. The phase-separated regime is due to strong dipole-dipole and dipole-monopole attractions that overcome the monopole-monopole repulsions, as expected based on previous work that focused on the solution behavior of dipolar molecules.14,35 Within the single-phase regime, a range of low ARDAll values was obtained, but no values below 20% were observed when all the TIS data sets at pH 7 were used. From inspection of individual-ARD plots (see Supp. Info.), the lowest TIS condition (10 mM) resulted in considerably higher ARD values and skewed the parameter sets (see below). Consequently, an additional ARD response surface was obtained by excluding the TIS = 10 mM (i.e., buffer only) condition, and that is shown in Figure 4B. The same two types of regimes (single phase and L-L split) are observed, but a portion of the ARD response surface clearly shows the desired low (< 10% ARD) behavior towards the lowest

μeff/Qeff values (bottom right corner). Figure 4C shows the comparison between predicted and experimental Rex profiles for the parameter ranges in the low-ARD region of Figure 4B, and this excludes a prediction for the 10 mM TIS conditions. Figure 4D is analogous to Figure 4C, but using the parameter range based on all TIS conditions for pH 7. The results in Figures 4C and 4D show that refining model parameters for individual TIS conditions is not needed for TIS values greater than 10 mM. However, predictions from low-c2 conditions are not accurate at high-c2 for strongly attractive conditions, even if the parameters are regressed only for those lowTIS conditions. This is discussed further below, within the context of Figure 5. Interestingly all parameter sets that provided good agreement between experimental and simulated Rex profiles in Figures 3 and 4 correspond to ARD regions below 20% for the B22 analysis (Figure 2). Therefore, with the exception of the lowest TIS conditions at pH 7, the parameters obtained from just low-c2 analysis (i.e., B22 values vs. TIS) provided quantitatively or


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semi-quantitatively accurate predictions of the high-c2 behavior. However, for very low TIS and strongly attractive electrostatic interactions at pH 7, there is poor agreement between the predictions and the experimental Rex data. This failure suggests that colloidal interaction models lack some of the key physics to properly describe the interactions at higher protein concentrations, and presumably that includes either accounting for higher terms in a multipole expansion, or working with a structurally more detailed PPI model. That suggests the need to use more detailed electrostatic models to accurately describe attractive conditions at low TIS and high-c2 conditions.9,14,19,22 Although doing so at high-c2 lies outside the scope of the present work, an approach for doing so at low-c2 is presented at the end of this report. There is an alternative interpretation of the results in Figure 4: predictions for electrostatic attractions at high concentrations are sensitive to the value of TIS at low TIS conditions. This would follow because the ion screening-length affects monopole-monopole repulsions, monopole-dipole and dipole-dipole attractions differently – each type of interaction decays as r-1, r-2 and r-3, respectively, with r denoting the intermolecular separation. For low TIS values, the balance between repulsion and attraction can be delicate, and strongly dependent on protein concentration. Therefore, small changes in TIS can tilt the balance towards a monopoledominated behavior or a dipole-dominated behavior as c2 increases. The Supporting Information illustrates this qualitatively for the interaction models in this work, and shows that strongly attractive PPI at high c2 and low TIS values below approximately 20 mM may be expected to be difficult to predict quantitatively using simple colloidal models. These concerns notwithstanding, the results in Figure 4 suggest that useful predictions about qualitative effects may be possible (e.g., very strong multi-body attractions) using simple models.


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The limitations at strongly attractive conditions is further illustrated in Figure 5, which shows the maximum TIS value at which a given set of μeff and Qeff values would result in L-L phase separation at pH 7 based on TMMC simulations, with the remaining model parameters fixed as described in the Methods. Figure 5 shows that high values of μeff/Qeff can result in phase separation with TIS as low as 50 mM. Increased net charge (Qeff) requires a higher minimum dipole moment (μeff) to observe L-L separation, illustrating the sensitive balance between the repulsive and attractive electrostatic interactions. As mentioned above, predictions might be improved at high-c2 conditions if one included higher multipoles. However, that would require more adjustable parameters and the accuracy of the model is still likely to break down once c2 reaches large enough values to be relevant to dense liquid phases (>> 100 mg/mL) because the detailed and complex surface-charge distribution would be expected to play a stronger role in producing localized multipole(s) on the protein surface.

Figure 5. Predicted phase separation at pH 7 as a function of dipole moment and TIS with increasing net charge values indicated by the arrow (Qeff = 3.4, 3.8 and 4.2). The colored area represents the conditions under which liquid-liquid phase separation was observed in the simulations (low TIS and higher µeff values with increasing Qeff).


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Higher resolution CG models To assess whether or not the anisotropic surface distribution of charged residues can predict the presence of multipole interactions for aCgn at pH 7, it was first considered whether an all-atom simulation of B22 would predict the experimental results. However, doing so requires explicit solvent molecules to be included at some level, otherwise the interaction energies will be grossly overestimated.17,61 As an alternative and computationally less expensive approach, PPI response surfaces were calculated by computing B22/B22,ST as a function of TIS and the ratio

εcc/εsr for εsr = 0.36 kBT and 0.93 kBT with the 1bAA and 4bAA models, respectively (see Methods section). The value of εsr was selected to assure that accurate B22 values would be obtained at high ionic strength for aCgn (see Supp. Info.). The response surface is expected to show one of three limiting cases, depending on the degree of anisotropy of the surface charge distribution: (1) monopole dominated behavior, such that B22 is large and positive at low TIS, and decreases monotonically with increasing TIS; (2) multipole dominated behavior, such that B22 is large and negative at low TIS, and increases monotonically with increasing TIS; (3) B22 shows a transition between monopole- and multipole-dominated regions as the ratio of εcc/εsr increases. This behavior can also be observed as a function of TIS. However, the experimental measurements had previously set the value for TIS, so the discussion below is developed from the perspective of εcc as a degree of freedom. The shape of the response surface should depend on the solution pH as well as the protein sequence and structure. Because these are CG models using implicit solvent, the effective value

εcc/εsr can be modified experimentally by changing the properties of the solution – e.g., by adding additional excipients that mediate PPI, as well as by specific-ion effects that lead to preferential exclusion or accumulation of ions near the protein surface.54,60,62 Figure 6A and 6B


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show the response surfaces for pH 5 and pH 7, respectively, using the 4bAA model and the published crystal structure of aCgn (PDB 1EX3)55. The two figures show that aCgn exhibits a transition from monopole-dominated to multipole-dominated behavior at both pH values (case 3, above). However, the change in pH shifts the εcc/εsr value at which this transition is located. For pH 5, the transition is observed at a very high value of the effective electrostatic interactions (εcc/εsr ~ 3), while it occurs at a much lower value (εcc/εsr ~ 2) for pH 7. Similar qualitative results were found by using the 1bAA model with small shifts in the εcc/εsr value under which this transition occurs (see Supp. Info.).

Figure 6. B22 response surfaces for the 4bAA CG molecular model as a function of TIS and relative strength of electrostatic vs. short-range attractive interactions (εcc/εsr) for: (A) pH 5 and (B) pH 7. The gray area in panel A corresponds to B22/B22,ST values above 5. (C) Experimental B22 vs TIS with best fit parameter sets (εcc/εsr = 0.97 for pH 5, εcc/εsr = 3.23 for pH 7). Open symbols, blue square (pH 5) and red circles (pH 7), represent the experimental data while lines and solid blue squares and red circles represent the simulated values from panels A and B, with lines drawn to guide the eye. The simulated values overlap with the experimental data except at the lowest TIS value for pH 7. The results are consistent with the existence of strong multipole interactions for pH 7 for aCgn, but not for pH 5. Figure 6C shows the comparison of experimental B22 vs TIS with the values predicted by the 4bAA model at each pH if one selects the value of εcc/εsr to minimize ARD (εcc/εsr = 0.97 for pH 5, εcc/εsr = 3.23 for pH 7). The value of εcc can be related to the ratio of the effective charge to the idealized value (ψi, see eq 10) of each charged residue due to ion binding or accumulation (see above). ψi is equal to 0.79 for pH 5, and 1.45 for pH 7 for the


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4bAA model results in Figure 6C (σi = 5 Å in eq 10). Physically, a ψi value different from 1 can be expected if there is non-ideal ion accumulation that causes greater than ideal charge screening ( < 1, as in pH 5) or multivalent ion (de)clouding and binding which can cause an increase in the effective charge (ψi > 1, as in pH 7)22,63. An alternative reason is simply that the 4bAA model groups entire side chains into single beads, and the bead diameter is different than that of the atom center that corresponds to the real charge site for a given amino acid. Therefore, caution should be used to not overinterpret the physical meaning of ψi > 1 with coarse-grained models. A minimum for B22 at low- to intermediate- TIS values for pH 7 is characteristic of strong electrostatic attractions caused by multipole interactions which, in theory, can be overcome by very long-ranged monopole repulsions at very low TIS. However, the experimental and simulated data show qualitative deviations at the lowest TIS values. This is possibly due to the inherent limitations of treating ion-screening with a mean-field description at such low ionic strength.63 The results in Figure 6 suggest that this level of CG structural model can accurately capture the anisotropic charge distribution of proteins in solution at low c2, but the model accuracy is lost at extremely low TIS (below 15 mM) and the parameter set may depend on pH, in contrast to previous work and canonical assumptions.9 This highlights the importance of experimental “training sets” to make even the higher-resolution CG models more effective. Finally, this type of response surface has the potential to be used more generally as a tool to assess how anisotropic surface-charge distributions affect net PPI without arbitrary definitions of charge “patches” or other geometric measures of anisotropic interactions that are difficult to generalize.19,20,64

Model-free correlation between high and low-c2 PPIs


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Finally, as experimental data were collected both at low and high protein concentrations for all solution conditions, it was of interest to consider whether the measured interaction parameters at low-c2 (i.e., B22) are qualitatively or quantitatively predictive of the experimentally determined PPI at high-c2, in terms of Sq=0 at a given choice of c2. This is illustrated in Figure 7, where Sq=0 is plotted as a function of the normalized B22 value at various concentrations. As expected, at sufficiently low protein concentration the structure factor value is approximately 1 for all B22 values, but as protein concentration increases (indicated by the arrows) the increase (decrease) of Sq=0 is more pronounced as B22 moves to net-attractive (net-repulsive) values. Additionally, the relationship between low and high PPI parameters is evident when one subtracts the steric contribution to Sq=0 at each concentrations as shown in Figure 7B. This shows that at a given protein concentration, the structure factor decreases with increasing B22 and illustrates that low-c2 PPIs are at least qualitatively predictive of high-c2 behavior for aCgn even if one does not utilize colloidal models to interpret or predict the high-c2 interactions. Not surprisingly, B22 is not quantitatively predictive of the interactions at high c2.

Figure 7. (A) Experimentally determined Sq=0 and (B) Experimental Sq=0 relative to that for steric-only interaction Sq=0,ST for aCgn at pH 5 and pH 7 as a function of normalized B22 value attained from low concentration fits. Arrows indicate increasing protein concentration. Each set of symbols corresponds to a different protein concentration, and B22/B22,ST is altered for each data


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set by changing TIS and/or pH. The protein concentrations are 1 (black rectangle), 10 (red circle), 30 (blue up-side triangle), 50 (green down-side triangle), 100 mg/ml (orange diamond).

SUMMARY AND CONCLUSIONS Zero-q static light scattering (SLS) was used to quantify net protein-protein interactions (PPI) of aCgn at a range of pH, ionic strength, and protein concentrations for conditions that produce netrepulsive or net-attractive electrostatic PPI at low ionic strengths. Simple colloidal models were tested for their ability to capture the net PPI from low- to high-c2 conditions for both netrepulsive and net-attractive interactions. This was done for two purposes. The first was to assess whether canonical colloidal interaction models could reasonably describe PPI as the ionic strength changes at a given pH for low-c2. The results show that simple colloidal models were able to quantitatively capture the data if a combination of screened monopole, screened dipole model, and short-ranged attractions (via a square-well model) were used. The second goal was to test whether the models and parameters from low-c2 data were predictive of high-c2 behavior. This used Transition Matrix Monte Carlo simulations as a natural method to predict the osmotic compressibility and therefore the excess Rayleigh scattering as a function of c2. The results show that the low-c2 model parameters are quantitatively predictive of the interactions at high concentrations if the net PPI are repulsive or slightly attractive compared to steric-only interactions. However, for strongly attractive interactions, where the effect of charge anisotropy is dominant, the low-c2 experimental data coupled with simple colloidal models was only able to qualitatively or semi-quantitatively predict the high-c2 behavior. Independent of the colloidal models, an approach was demonstrated for identifying when anisotropic charge distributions are expected to dominate net PPI as a function of ionic strength, without the need for assuming or defining charge “patches” or other arbitrary measure of surface anisotropy.


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ASSOCIATED CONTENT Supporting Information. The supporting information includes: detailed equations for the electrostatic potential of mean force models; apparent molecular weight values from low-c2 fit for aCgn at pH 7 and 5 as function of TIS; ARD surface response plots for individual TIS values; B22 as a function of εsr for the 1bAA and 4bAA CG molecular models and mapping of B22 values for 1bAA CG model. This material is available free of charge on the ACS Publication website.

AUTHOR INFORMATION Corresponding Author * EMF: Tel 302-831-0102, fax 302-831-1048, e-mail [email protected]; CJR: Tel 302-831-0838, fax 302-831-1048, e-mail [email protected] ACKNOWLEDGMENT Support from the Biomolecular Interaction Technologies Center (BITC), the National Institutes of Health (R01 EB006006) and the National Science Foundation (NSF GRFP to M.A. Woldeyes) is gratefully acknowledged. Dr. R. A. Curtis is also acknowledged for many helpful discussions.

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TOC Graphic


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Figure 1A 215x166mm (300 x 300 DPI)


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Figure 1B 279x215mm (300 x 300 DPI)





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Figure 3C 215x166mm (300 x 300 DPI)


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Figure 4D 215x166mm (300 x 300 DPI)


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Figure 5 279x215mm (300 x 300 DPI)



Fig 6A. See caption in file 215x166mm (300 x 300 DPI)


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Fig 6B. See caption in file 215x166mm (300 x 300 DPI)





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